# find the Laplace transform by the method of the unit step function. f(t)={(t , 0\leq t <1),(1 , t\geq 1):}

find the Laplace transform by the method of the unit step function
$\left(t\right)=\left\{\begin{array}{ll}t& 0\le t<1\\ 1& t\ge 1\end{array}$

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Step 1
The given function,
$\left(t\right)=\left\{\begin{array}{ll}t& 0\le t<1\\ 1& t\ge 1\end{array}$
Step 2
Obtain $Lf\left(t\right)$ as follows.
$F\left(s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}\left(f\left(t\right)\right)$
$={\int }_{0}^{1}{e}^{-st}\left(t\right)dt+{\int }_{1}^{\mathrm{\infty }}{e}^{-st}\left(1\right)dt$
$=\frac{-s{e}^{-s}-{e}^{-s}+1}{{s}^{2}}+\frac{{e}^{-s}}{s}$
$=\frac{-{s}^{2}{e}^{-s}-s{e}^{-s}+s+{s}^{2}{e}^{-s}}{{s}^{3}}$
$=\frac{-s{e}^{-s}+s}{{s}^{3}}$
$=\frac{-{e}^{-s}+1}{{s}^{2}}$
Thus , the Laplace transform by the method of the unit step function.